Harmonic balance method

An approximate method for the study of non-linear oscillating systems described by ordinary non-linear differential equations. The essence of the method is to replace the non-linear forces in the oscillating systems by specially-constructed linear functions, so that the theory of linear differential equations may be employed to find approximate solutions of the non-linear systems.

The linear functions are constructed by a special method, known as harmonic linearization. Let the given non-linear function (force) be

$$ F ( x, \dot{x} ) \equiv \ \epsilon f ( x, \dot{x} ),\ \ \dot{x} = \frac{dx }{dt } , $$

where $ \epsilon $ is a small parameter. Harmonic linearization is the replacement of $ F ( x, \dot{x} ) $ by the linear function

$$ F _ {l} ( x, \dot{x} ) = \ kx + \lambda \dot{x} , $$

where the parameters $ k $ and $ \lambda $ are computed by the formulas

$$ k ( a) = \ { \frac \epsilon {\pi a } } \int\limits _ { 0 } ^ { {2 } \pi } f ( a \cos \psi , - a \omega \sin \psi ) \cos \psi d \psi , $$

$$ \lambda ( a) = - { \frac \epsilon {\pi a \omega } } \int\limits _ { 0 } ^ { {2 } \pi } f ( a \ \cos \psi , - a \omega \sin \psi ) \sin \psi d \psi , $$

$$ \psi = \omega t + \theta . $$

If $ x = a \cos ( \omega t + \theta ) $, $ a = \textrm{ const } $, $ \omega = \textrm{ const } $, $ \theta = \textrm{ const } $, the non-linear force $ F( x, \dot{x} ) $ is a periodic function of time, and its Fourier series expansion contains, generally speaking, an infinite number of harmonics, having the frequencies $ n \omega $, $ n = 1, 2 \dots $ i.e. it is of the form

$$ \tag{1 } F ( x, \dot{x} ) = \ \sum _ {n = 0 } ^ \infty F _ {n} \cos ( n \omega t + \theta _ {n} ). $$

The term $ F _ {1} \cos ( \omega t + \theta _ {1} ) $ is called the fundamental harmonic of the expansion (1). The amplitude and the phase of the linear function $ F _ {l} $ coincide with the respective characteristics of the fundamental harmonic of the non-linear force.

For the differential equation

$$ \tag{2 } \dot{x} dot + \omega ^ {2} x + F ( x, \dot{x} ) = 0, $$

which is typical in the theory of quasi-linear oscillations, the harmonic balance method consists in replacing $ F( x, \dot{x} ) $ by the linear function $ F _ {l} $; instead of equation (2), one then considers the equation

$$ \tag{3 } \dot{x} dot + \lambda \dot{x} + k _ {1} x = 0, $$

where $ k _ {1} = \omega ^ {2} + k $. It is usual to call $ F _ {l} $ the equivalent linear force, $ \lambda $ the equivalent damping coefficient and $ k _ {1} $ the equivalent elasticity coefficient. It has been proved that if the non-linear equation (2) has a solution of the form

$$ x = a \cos ( \omega t + \theta ), $$

where

$$ \dot{a} = O ( \epsilon ),\ \ \dot \omega = O ( \epsilon ), $$

then the order of the difference between the solutions of (2) and (3) is $ \epsilon ^ {2} $. In the harmonic balance method the frequency of the oscillation depends on the amplitude $ a $( through the quantities $ k $ and $ \lambda $).

The harmonic balance method is used to find periodic and quasi-periodic oscillations, periodic and quasi-periodic conditions in automatic control theory, as well as stationary conditions, and in the studies of their stability. It is extensively used in automatic control theory.

References

[1]	N.M. Krylov, N.N. Bogolyubov, "Introduction to non-linear mechanics" , Princeton Univ. Press (1947) (Translated from Russian)
[2]	N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian)
[3]	E.P. Popov, I.P. Pal'tov, "Approximate methods for studying non-linear automatic systems" , Translation Services , Ohio (1963) (Translated from Russian)